# American Institute of Mathematical Sciences

July  2017, 22(5): 1965-1975. doi: 10.3934/dcdsb.2017115

## Topological stability in set-valued dynamics

 1 Instituto de Matemàtica y Ciencias Afines (IMCA), Universidad Nacional de Ingeniera Calle Los Biòlogos 245, 15012 Lima, Perù 2 Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530,21945-970 Rio de Janeiro, Brazil 3 Institut de Mathématiques Université de Bordeaux Ⅰ, 33405, Talence, France

* Corresponding author: Carlos Arnoldo Morales Rojas

Received  November 2016 Revised  January 2017 Published  March 2017

Fund Project: Work partially supported by CNPq from Brazil and MATHAMSUD 15 MATH05-ERGOPTIM, Ergodic Optimization of Lyapunov Exponents.

We propose a definition of topological stability for set-valued maps. We prove that a single-valued map which is topologically stable in the set-valued sense is topologically stable in the classical sense [14]. Next, we prove that every upper semicontinuous closed-valued map which is positively expansive [15] and satisfies the positive pseudo-orbit tracing property [9] is topologically stable. Finally, we prove that every topologically stable set-valued map of a compact metric space has the positive pseudo-orbit tracing property and the periodic points are dense in the nonwandering set. These results extend the classical single-valued ones in [1] and [14].

Citation: Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115
##### References:
 [1] N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances, North-Holland Mathematical Library, 52. North-Holland Publishing Co. , Amsterdam, 1994. Google Scholar [2] J. -P. Aubin and H. Frankowska, Set-valued Analysis, Reprint of the 1990 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc. , Boston, MA, 2009. Google Scholar [3] R. Bowen, ω-limit sets for axiom A diffeomorphisms, J. Differential Equations, 18 (1975), 333-339.  doi: 10.1016/0022-0396(75)90065-0.  Google Scholar [4] D. Carrasco-Olivera, A. R. Metzger and C. A. Morales, Logarithmic expansion, entropy and dimension for set-valued maps, Preprint, (2016), to appear. Google Scholar [5] D. Carrasco-Olivera, R. Metzger Alvan and C. A. Morales, Topological entropy for set-valued maps, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3461-3474.  doi: 10.3934/dcdsb.2015.20.3461.  Google Scholar [6] W. Cordeiro and M. J. Pacifico, Continuum-wise expansiveness and specification for set-valued functions and topological entropy, Proc. Amer. Math. Soc., 144 (2016), 4261-4271.  doi: 10.1090/proc/13168.  Google Scholar [7] M. Eisenberg, Expansive transformation semigroups of endomorphisms, Fund. Math., 59 (1966), 313-321.   Google Scholar [8] J. P. Kelly and T. Tennant, Topological entropy for set-valued functions, arXiv: 1509.08413. Google Scholar [9] S. Y. Pilyugin and J. Rieger, Shadowing and inverse shadowing in set-valued dynamical systems. Contractive case, Topol. Methods Nonlinear Anal., 32 (2008), 139-149.   Google Scholar [10] S. Y. Pilyugin, Shadowing in Dynamical Systems Lecture Notes in Mathematics, 1706. Springer-Verlag, Berlin, 1999. Google Scholar [11] B. E. Raines and T. Tennant, The specification property on a set-valued map and its inverse limit, arXiv: 1509.08415. Google Scholar [12] W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc., 1 (1950), 769-774.  doi: 10.1090/S0002-9939-1950-0038022-3.  Google Scholar [13] P. Walters, On the pseudo-orbit tracing property and its relationship to stability. The structure of attractors in dynamical systems (Proc. Conf. , North Dakota State Univ. , Fargo, N. D. , 1977), Lecture Notes in Math. , Springer, Berlin, 668 (1978), 231-244. Google Scholar [14] P. Walters, Anosov diffeomorphisms are topologically stable, Topology, 9 (1970), 71-78.  doi: 10.1016/0040-9383(70)90051-0.  Google Scholar [15] R. K. Williams, A note on expansive mappings, Proc. Amer. Math. Soc., 22 (1969), 145-147.  doi: 10.1090/S0002-9939-1969-0242143-4.  Google Scholar

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##### References:
 [1] N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances, North-Holland Mathematical Library, 52. North-Holland Publishing Co. , Amsterdam, 1994. Google Scholar [2] J. -P. Aubin and H. Frankowska, Set-valued Analysis, Reprint of the 1990 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc. , Boston, MA, 2009. Google Scholar [3] R. Bowen, ω-limit sets for axiom A diffeomorphisms, J. Differential Equations, 18 (1975), 333-339.  doi: 10.1016/0022-0396(75)90065-0.  Google Scholar [4] D. Carrasco-Olivera, A. R. Metzger and C. A. Morales, Logarithmic expansion, entropy and dimension for set-valued maps, Preprint, (2016), to appear. Google Scholar [5] D. Carrasco-Olivera, R. Metzger Alvan and C. A. Morales, Topological entropy for set-valued maps, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3461-3474.  doi: 10.3934/dcdsb.2015.20.3461.  Google Scholar [6] W. Cordeiro and M. J. Pacifico, Continuum-wise expansiveness and specification for set-valued functions and topological entropy, Proc. Amer. Math. Soc., 144 (2016), 4261-4271.  doi: 10.1090/proc/13168.  Google Scholar [7] M. Eisenberg, Expansive transformation semigroups of endomorphisms, Fund. Math., 59 (1966), 313-321.   Google Scholar [8] J. P. Kelly and T. Tennant, Topological entropy for set-valued functions, arXiv: 1509.08413. Google Scholar [9] S. Y. Pilyugin and J. Rieger, Shadowing and inverse shadowing in set-valued dynamical systems. Contractive case, Topol. Methods Nonlinear Anal., 32 (2008), 139-149.   Google Scholar [10] S. Y. Pilyugin, Shadowing in Dynamical Systems Lecture Notes in Mathematics, 1706. Springer-Verlag, Berlin, 1999. Google Scholar [11] B. E. Raines and T. Tennant, The specification property on a set-valued map and its inverse limit, arXiv: 1509.08415. Google Scholar [12] W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc., 1 (1950), 769-774.  doi: 10.1090/S0002-9939-1950-0038022-3.  Google Scholar [13] P. Walters, On the pseudo-orbit tracing property and its relationship to stability. The structure of attractors in dynamical systems (Proc. Conf. , North Dakota State Univ. , Fargo, N. D. , 1977), Lecture Notes in Math. , Springer, Berlin, 668 (1978), 231-244. Google Scholar [14] P. Walters, Anosov diffeomorphisms are topologically stable, Topology, 9 (1970), 71-78.  doi: 10.1016/0040-9383(70)90051-0.  Google Scholar [15] R. K. Williams, A note on expansive mappings, Proc. Amer. Math. Soc., 22 (1969), 145-147.  doi: 10.1090/S0002-9939-1969-0242143-4.  Google Scholar
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